Elliptical distributions

Generalising multivariate Gaussians to anything that has a density function of the form $$f(x)\propto g((x-\mu {)}^{\prime }{\mathrm{\Sigma }}^{-1}(x-\mu ))$$ where $\mu$ is the mean vector, $\mathrm{\Sigma }$ is a positive definite matrix, and $g:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$. In fact, we do not need the density function to exist; it’s ok if $\mathrm{\Sigma }$ is positive semi-definite or to allow $g$ to be a generalised function.

Baby steps, though; let us have densities for now. If the mean of such an $X\sim f$ RV exists, it is $\mu$, and $\mathrm{\Sigma }$ is proportional to the covariance matrix of $X$, if such a covariance matrix exists.

I assume they did not invent this idea, but Davison and Ortiz (2019) points out that if you have a least-squares-compatible model, usually it can generalize to any elliptical density, which includes many M-estimator-style robust losses.